Average Error: 13.4 → 1.2
Time: 4.8s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\frac{\frac{x}{wj \cdot wj - 1} \cdot \left(wj - 1\right)}{e^{wj}} + \left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\frac{\frac{x}{wj \cdot wj - 1} \cdot \left(wj - 1\right)}{e^{wj}} + \left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)
double f(double wj, double x) {
        double r198275 = wj;
        double r198276 = exp(r198275);
        double r198277 = r198275 * r198276;
        double r198278 = x;
        double r198279 = r198277 - r198278;
        double r198280 = r198276 + r198277;
        double r198281 = r198279 / r198280;
        double r198282 = r198275 - r198281;
        return r198282;
}
↓
double f(double wj, double x) {
        double r198283 = x;
        double r198284 = wj;
        double r198285 = r198284 * r198284;
        double r198286 = 1.0;
        double r198287 = r198285 - r198286;
        double r198288 = r198283 / r198287;
        double r198289 = r198284 - r198286;
        double r198290 = r198288 * r198289;
        double r198291 = exp(r198284);
        double r198292 = r198290 / r198291;
        double r198293 = 4.0;
        double r198294 = pow(r198284, r198293);
        double r198295 = 2.0;
        double r198296 = pow(r198284, r198295);
        double r198297 = r198294 + r198296;
        double r198298 = 3.0;
        double r198299 = pow(r198284, r198298);
        double r198300 = r198297 - r198299;
        double r198301 = r198292 + r198300;
        return r198301;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.4
Target12.7
Herbie1.2
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.4

    \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]

  2. Simplified12.7

    \[\leadsto \color{blue}{\left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) - \frac{wj}{wj + 1}}\]
  3. Using strategy rm

  4. Applied associate--l+6.9

    \[\leadsto \color{blue}{\frac{\frac{x}{wj + 1}}{e^{wj}} + \left(wj - \frac{wj}{wj + 1}\right)}\]

  5. Taylor expanded around 0 1.2

    \[\leadsto \frac{\frac{x}{wj + 1}}{e^{wj}} + \color{blue}{\left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)}\]
  6. Using strategy rm

  7. Applied flip-+1.2

    \[\leadsto \frac{\frac{x}{\color{blue}{\frac{wj \cdot wj - 1 \cdot 1}{wj - 1}}}}{e^{wj}} + \left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)\]

  8. Applied associate-/r/1.2

    \[\leadsto \frac{\color{blue}{\frac{x}{wj \cdot wj - 1 \cdot 1} \cdot \left(wj - 1\right)}}{e^{wj}} + \left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)\]

  9. Simplified1.2

    \[\leadsto \frac{\color{blue}{\frac{x}{wj \cdot wj - 1}} \cdot \left(wj - 1\right)}{e^{wj}} + \left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)\]

  10. Final simplification1.2

    \[\leadsto \frac{\frac{x}{wj \cdot wj - 1} \cdot \left(wj - 1\right)}{e^{wj}} + \left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)\]

Reproduce

herbie shell --seed 2020027 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

  :herbie-target
  (- wj (- (/ wj (+ wj 1)) (/ x (+ (exp wj) (* wj (exp wj))))))

  (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))